# Protein Stability and Delta Delta G (ddG): What Mutation Stability Values Mean

Protein stability is a thermodynamic property, not a structural one. When a folded protein sits in solution, it exists in an equilibrium between the native state (N) and the unfolded state (U), and the position of that equilibrium is set by the Gibbs free energy of unfolding. A mutation changes that free energy, and the size of the change is what ddG measures. If you work with proteins, you will meet ddG in mutagenesis studies, in stability engineering, in biophysical characterization reports, and in the output of prediction tools.

The number itself is small and easy to misread. A ddG of -1.5 kcal/mol sounds trivial, yet it can shift the unfolded population by more than tenfold. That gap between the size of the number and the size of its effect is where most confusion starts. This article covers what ddG is, how it is calculated from experimental data, how to read it correctly, and where the underlying assumptions stop holding.

## Quick Answer

- Protein stability is expressed as the Gibbs free energy of unfolding, ΔG(unfolding) = G(unfolded) - G(native), for the equilibrium N <-> U. A positive value means the native state is favored [7].
- ddG (ΔΔG) is the difference in unfolding free energy between a variant and the wild type, usually in kcal/mol. Its sign tells you whether the mutation destabilizes or stabilizes the protein [7].
- The two-state relation is $K = \exp\left(-\frac{\Delta G}{RT}\right)$ where K = [U]/[N], R is the gas constant and T is the absolute temperature. The fraction unfolded is f_U = K/(1 + K).
- At 25 C, RT = 0.5925 kcal/mol, so each 1 kcal/mol of ddG changes the folding equilibrium constant about 5.4-fold, and 1.364 kcal/mol changes it 10-fold.
- Sign conventions differ between sources. In the experimental unfolding convention, ddG = ΔG(unfolding, mutant) - ΔG(unfolding, wild type), so negative means destabilizing [2]. FoldX reports the opposite: positive means destabilizing [8].
- Always check the convention, temperature, pH and denaturant before comparing ddG values from different tools or papers.

## What Protein Stability Actually Means

A folded protein is only marginally more stable than its unfolded state. Reported free energies of folding for small, mostly single-domain proteins fall within about -1 to -20 kcal/mol in the folding direction, with a distribution peaking near -5 kcal/mol. The range most often quoted in the literature is 5 to 15 kcal/mol, read as "proteins are marginally stable" [3]. That is a small margin. A handful of hydrogen bonds or a few buried hydrophobic side chains can account for it.

The equilibrium is written as N <-> U, and the unfolding equilibrium constant is K = [U]/[N]. The standard relation between K and free energy is the same one used for any equilibrium constant, K° = exp(-ΔrG°/RT) [1]. Rearranged for unfolding, this gives K = exp(-ΔG/RT), where ΔG is the unfolding free energy, R is the gas constant and T is the temperature in kelvin. The fraction of protein that is unfolded follows directly: f_U = K/(1 + K).

The intuition is straightforward. A large positive ΔG makes the exponent strongly negative, so K is tiny and almost everything is folded. A ΔG near zero makes K near 1, so half the molecules are unfolded. Because the relation is exponential, small changes in ΔG produce large changes in the unfolded fraction. That is the entire reason ddG matters.

Two-state behavior means only two conformations are populated in solution, the native and the fully unfolded protein [5]. Real proteins often fold through intermediates, and the two-state model is an approximation. It works best for small proteins with unfolding enthalpies of about 50 to 200 kcal/mol. Larger proteins such as antibodies, with unfolding enthalpies near 1000 kcal/mol, usually need multistate or cooperative models [5].

## How ddG Is Measured and Calculated

### Chemical denaturation

The standard approach is the linear extrapolation method. A protein is titrated with urea or guanidine hydrochloride, and ΔG is plotted against denaturant concentration. The slope of that line is the m value, and the intercept at 0 M denaturant is ΔG(H2O), the stability in water [2].

$$\Delta G(H_2O) = m \times [\text{urea}]_{1/2}$$

At the midpoint of the transition, [urea]1/2, ΔG = 0, so the product of the m value and the midpoint concentration recovers the stability. For VlsE, a 341-residue protein, m = 3865 cal mol^-1 M^-1 and [urea]1/2 = 1.19 M, giving 4.6 kcal/mol, which matches the reported value [2]. The 36-residue villin headpiece subdomain VHP has a reported ΔG(H2O) of 2.7 kcal/mol under the same conditions [2].

Denaturant m values correlate strongly with the change in solvent-accessible surface area on unfolding: R = 0.84 for urea and 0.87 for guanidine hydrochloride, rising to 0.90 when disulfide effects are included [4]. For proteins with a simple two-state mechanism, the amount of surface exposed on unfolding is the main structural determinant of both the m value and the heat capacity change ΔCp [4].

To estimate ddG for a mutant from chemical denaturation, Pace et al. use the change in midpoint concentration multiplied by the average m value of wild type and variants [2]. For the VlsE variant I128V, Δ[urea]1/2 = -0.31 M and the average m is 3590 cal/mol/M, giving -1.11 kcal/mol, reported as -1.1 kcal/mol [2].

### Thermal denaturation

The melting temperature Tm is the midpoint of the thermal unfolding curve, where native and unfolded states are equally populated and ΔG = 0 in a two-state model [2]. For hen lysozyme, differential scanning calorimetry gives Tm = 62 C, a heat capacity increase on unfolding of 2.27 kcal mol^-1 K^-1 and a total unfolding enthalpy of 138 kcal/mol under the conditions of Seelig and Seelig [5].

From thermal data, ddG is estimated as ΔTm multiplied by the average entropy of unfolding at Tm (ΔSm) [2]. As a conversion example, ΔTm = -5 K with ΔSm = 71 cal/mol/K gives ddG of about -0.36 kcal/mol.

### Thermal shift assays

Differential scanning fluorimetry, also called a thermal shift assay, measures the temperature at which a protein unfolds by tracking the rising fluorescence of a dye that binds exposed hydrophobic regions. The shift in midpoint (ΔTm) with a ligand relates to ligand binding affinity, and the assay runs on a real-time PCR instrument [6]. It is a screening method, not a direct measurement of ΔG.

### Prediction tools

FoldX is an empirical force field method that predicts the effect of a single-point variation from a linear combination of empirical free energy terms computed on a 3D structure [7]. Rosetta estimates ddG from the difference in Rosetta energy between modeled wild-type and mutant structures [7]. Both are widely used and both carry meaningful error.

## Worked Example

Take a two-state unfolding equilibrium N <-> U at 25 C, so T = 298.15 K. With R = 8.314462618 J/mol/K divided by 4184 J/kcal, R = 1.98720e-3 kcal/mol/K, and RT = 0.5925 kcal/mol.

Wild type has ΔG(unfolding) = 5.0 kcal/mol. Then:

$$K = \exp\left(-\frac{5.0}{0.5925}\right) = \exp(-8.439) = 2.163 \times 10^{-4}$$

$$f_U = \frac{K}{1 + K} = 2.162 \times 10^{-4}$$

That is 0.0216% unfolded, or roughly one unfolded molecule per 4,624 folded.

Now introduce a mutation that destabilizes the protein by 1.5 kcal/mol, leaving ΔG = 3.5 kcal/mol:

$$K = \exp\left(-\frac{3.5}{0.5925}\right) = 2.719 \times 10^{-3}$$

$$f_U = \frac{K}{1 + K} = 2.712 \times 10^{-3}$$

That is 0.271% unfolded, or one unfolded molecule per 368 folded. The ratio of unfolded populations is 12.54, which matches exp(1.5/RT) = 12.57 to within the 1 + K correction.

In the unfolding convention (ΔG mutant minus ΔG wild type), this mutation has ddG = 3.5 - 5.0 = -1.5 kcal/mol, negative meaning destabilizing, as in the Pace et al. tables [2]. In the FoldX-style folding convention, the same mutation is reported as +1.5 kcal/mol, positive meaning destabilizing [8].

For scale, other reference points at 25 C: ΔG = 10 kcal/mol gives f_U = 4.7e-8, ΔG = 15 kcal/mol gives 1.0e-11, ΔG = 1.0 kcal/mol gives 0.156, and ΔG = 0 gives 0.5. A 5 kcal/mol protein can lose several kcal/mol before a large unfolded fraction appears.

## Reading ddG Correctly

The sign convention is the first thing to check. In the experimental unfolding convention, ddG = ΔG(unfolding, mutant) - ΔG(unfolding, wild type), so negative values indicate a decrease in stability [2]. FoldX notes that the ddG sign is only a convention and that, thermodynamically, a negative ddG means the system releases energy and reaches a more stable state, so in FoldX output positive ddG means destabilizing [8]. Pancotti et al. report predictor bias with destabilizing variants as negative values, the unfolding convention, which is opposite to FoldX output [7][8].

ddG is antisymmetric. The reverse mutation, mutant back to wild type, must have a ddG of equal magnitude and opposite sign [7]. If a tool violates this, its numbers are not internally consistent.

| Convention | Destabilizing | Stabilizing | Used by |
|---|---|---|---|
| Unfolding (ΔG mut - ΔG wt) | negative | positive | Pace et al. tables, Pancotti et al. benchmark [2][7] |
| Folding (FoldX output) | positive | negative | FoldX [8] |

Benchmark performance gives a realistic sense of accuracy. On S669, a set of 669 variants from ThermoMutDB not in common training sets, Pearson correlations of 21 predictors were 0.21 to 0.5 for direct variants and 0 to 0.45 for reverse variants, rising to 0.51 to 0.62 for antisymmetric methods on the combined set [7]. On the combined set, FoldX had r = 0.31, RMSE 2.39 kcal/mol and MAE 1.53 kcal/mol, while Rosetta had r = 0.47, RMSE 2.69 and MAE 2.05. The best tools reached about r = 0.6 with RMSE near 1.5 and MAE near 1.05 to 1.1 [7].

Predictors tend to compress ddG predictions toward zero, most are biased toward destabilizing predictions because training data are dominated by destabilizing variants, and strongly stabilizing variants are the hardest to predict [7].

## Common Mistakes

- **Comparing ddG values without checking the sign convention.** A -1.5 kcal/mol from one source and a +1.5 kcal/mol from another may describe the same mutation. Check the convention, temperature, pH and denaturant before comparing numbers.
- **Treating a ΔTm from a thermal shift assay as a ddG.** The conversion ddG ≈ ΔTm x ΔSm assumes two-state behavior and requires an entropy term. A raw ΔTm is a temperature shift, not an energy.
- **Assuming a small ddG is negligible.** At 25 C, 1 kcal/mol changes the folding equilibrium constant about 5.4-fold. A 1.5 kcal/mol destabilization raises the unfolded population more than twelvefold.
- **Ignoring the temperature dependence of ΔG.** Stability depends on temperature, pH and buffer. A ddG measured at one set of conditions does not transfer directly to another.
- **Trusting a single predictor's absolute value.** Benchmark RMSE values near 1.5 to 2.7 kcal/mol mean individual predictions carry substantial error. Use predictions for ranking and screening, then confirm experimentally.
- **Forgetting antisymmetry.** If a tool gives different magnitudes for a mutation and its reverse, the output is not thermodynamically consistent.

## Limitations

Two-state analysis requires reversible unfolding without populated intermediates. ΔG from thermal and chemical denaturation can differ, and ΔG depends on temperature, pH and buffer. The 5 to 15 kcal/mol range is commonly quoted, but Sorokina et al. describe it as such while themselves reporting a distribution from about 1 to 20 kcal/mol peaking near 5 [3].

Seelig and Seelig argue that the standard chemical-equilibrium two-state model mispredicts calorimetric temperature profiles and that free energy alone is not a good criterion of stability [5]. This is a minority view, but it is a useful reminder that the model is a model. Sorokina et al. also argue against parts of the standard picture of spontaneous folding [3].

The FoldX convention statement comes from the FoldX website FAQ page, not from a peer-reviewed paper [8]. Benchmark error values are dataset specific to S669; other benchmarks give different numbers, and tool versions change. The ProTherm entry count of 17,113 entries from 771 proteins across 1,497 articles is from the 2006 release paper, and the database's current status and size were not verified [10]. ThermoMutDB is a manually curated database with over 14,669 experimental thermodynamic data points for wild-type and mutant proteins, including unfolding Gibbs free energy and melting temperature changes, with a RESTful API [9].

## Frequently Asked Questions

### What is ddG protein stability?

ddG, written ΔΔG, is the difference in unfolding free energy between a variant and the wild type, usually reported in kcal/mol. It quantifies how much a mutation shifts the N <-> U equilibrium. The sign tells you whether the variant is destabilizing or stabilizing, but only after you confirm which convention the source uses.

### How do I convert a melting temperature shift into a ddG?

Multiply the ΔTm by the average entropy of unfolding at Tm (ΔSm): ddG ≈ ΔTm x ΔSm [2]. This assumes two-state behavior. For example, ΔTm = -5 K with ΔSm = 71 cal/mol/K gives about -0.36 kcal/mol. Without a measured ΔSm, the conversion is an estimate, not a measurement.

### How accurate is ddG prediction?

On the S669 benchmark, Pearson correlations for 21 predictors ranged from 0.21 to 0.5 for direct variants and 0 to 0.45 for reverse variants, with antisymmetric methods reaching 0.51 to 0.62 on the combined set [7]. FoldX had r = 0.31 and RMSE 2.39 kcal/mol; Rosetta had r = 0.47 and RMSE 2.69. The best tools reached about r = 0.6 with RMSE near 1.5 kcal/mol. Predictions are useful for ranking, not for absolute values.

### What does a stabilizing mutation look like in ddG terms?

A stabilizing mutation increases the unfolding free energy, so in the unfolding convention its ddG is positive. In FoldX output, the same mutation appears as a negative ddG [8]. Stabilizing variants are the hardest class for predictors to get right because training data are dominated by destabilizing variants [7].

### Why is the delta G of unfolding so small for most proteins?

Reported folding free energies for small, mostly single-domain proteins fall within about -1 to -20 kcal/mol, peaking near -5 kcal/mol, and the commonly quoted range is 5 to 15 kcal/mol [3]. Proteins are marginally stable, which is enough because the exponential relation between ΔG and K means even a few kcal/mol keeps the unfolded fraction very low.

## References

1. [IUPAC Gold Book: standard equilibrium constant](https://goldbook.iupac.org/terms/view/S05915)
2. [Pace et al. 2011, Contribution of hydrophobic interactions to protein stability, J Mol Biol](https://doi.org/10.1016/j.jmb.2011.02.053)
3. [Sorokina et al. 2022, Is protein folding a thermodynamically unfavorable, active, energy-dependent process?, Int J Mol Sci](https://doi.org/10.3390/ijms23010521)
4. [Myers, Pace and Scholtz 1995, Denaturant m values and heat capacity changes, Protein Sci](https://doi.org/10.1002/pro.5560041020)
5. [Seelig and Seelig 2023, Protein stability: analysis of heat and cold denaturation, J Phys Chem B](https://doi.org/10.1021/acs.jpcb.3c00882)
6. [Niesen, Berglund and Vedadi 2007, Differential scanning fluorimetry, Nat Protoc](https://doi.org/10.1038/nprot.2007.321)
7. [Pancotti et al. 2022, Predicting protein stability changes upon single-point mutation, Brief Bioinform](https://doi.org/10.1093/bib/bbab555)
8. [FoldX: How to explain the ΔΔG?](https://foldxsuite.crg.eu/node/789)
9. [Xavier et al. 2021, ThermoMutDB, Nucleic Acids Res](https://doi.org/10.1093/nar/gkaa925)
10. [Kumar et al. 2006, ProTherm and ProNIT, Nucleic Acids Res](https://doi.org/10.1093/nar/gkj103)

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